Subgroups — #89696

Order 448.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#1136	C₂	1	2	yes	maximal
3	C₄⋊(C₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₇	C₄	1	2	yes	—
5	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₄²×C₇	C₄	1	1	yes	characteristic
7	C₄²×C₇	C₄	1	1	yes	characteristic
8	C₄²×C₇	C₂²	1	2	yes	—
9	#1136	—	2	4	no	—
10	C₂×C₇⋊C₈	C₂²	1	4	yes	—
11	#423	—	7	1	no	maximal, Sylow
12	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₇	—	2	4	no	—
14	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₇	D₄	1	2	yes	—
18	C₂×C₄×C₇	Q₈	1	2	yes	—
19	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₇⋊C₈	—	2	4	no	—
21	C₂×M₁₆	—	7	2	no	—
22	C₂×C₄²	D₇	1	1	yes	characteristic
23	C₄⋊C₈	—	7	4	no	—
24	C₂²×C₇	—	2	2	no	—
25	C₂²×C₇	C₄⋊C₄	1	2	yes	—
26	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₇	—	2	2	no	—
28	C₄×C₇	C₄⋊C₄	1	2	yes	—
29	C₄×C₇	M₁₆	1	4	yes	complemented
30	C₄×C₇	C₂×D₄	1	1	yes	characteristic
31	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
32	C₂²×C₄	Dic₇	1	2	yes	—
33	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
34	M₁₆	—	14	4	no	—
35	C₂×C₈	—	7	4	no	—
36	C₄²	Dic₇	1	1	yes	characteristic
37	C₄²	Dic₇	1	1	yes	characteristic
38	C₄²	C₂×D₇	1	2	yes	—
39	C₂×C₇	—	2	2	no	—
40	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₇	C₂×M₁₆	1	2	yes	—
42	C₂³	C₂×Dic₇	1	1	yes	characteristic
43	C₂×C₄	—	2	4	no	—
44	C₂×C₄	Dic₁₄	1	2	yes	—
45	C₂×C₄	D₂₈	1	2	yes	—
46	C₂×C₄	C₂×Dic₇	1	2	yes	—
47	C₂×C₄	C₂×Dic₇	1	2	yes	—
48	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
49	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
50	C₈	—	14	4	no	—
51	C₇	#423	1	1	yes	characteristic, complemented, Sylow
52	C₂²	—	2	2	no	—
53	C₂²	C₄⋊Dic₇	1	2	yes	—
54	C₂²	C₂²×Dic₇	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	#1136	1	4	yes	complemented
57	C₄	C₄⋊Dic₇	1	2	yes	—
58	C₄	C₂×Dic₁₄	1	1	yes	characteristic
59	C₄	C₂×D₂₈	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×#1136	1	2	yes	—
62	C₂	C₂×C₄⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₄
Fitting	C₂×C₄²×C₇	C₂
prime radical	C₂³×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×C₄²	D₇
hypocenter	C₇	#423
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₄⋊(C₇⋊C₈)	C₂	semidirect
29. C₄×C₇	M₁₆	semidirect
51. C₇	#423	semidirect
56. C₄	#1136	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄²×C₇	2
C₂×#1136	2
C₄⋊(C₇⋊C₈)	2
#423	7